A linear-over-squared-pairwise sum bounds a reciprocal total

Prove

$\frac{x}{\left(y+z\right)^2}+\frac{y}{\left(z+x\right)^2}+\frac{z}{\left(x+y\right)^2} \geq \frac{9}{4\left(x+y+z\right)}$

for any three positive reals x, y, z.

Equivalently, prove

$\frac{s-a}{a^2}+\frac{s-b}{b^2}+\frac{s-c}{c^2} \geq \frac{9}{4s}$ ,

where a, b, c are the sides of a triangle and $s=\frac{a+b+c}{2}$ .

Equivalently, show

$\frac{\sin B \sin C}{\sin^2 \frac{A}{2}} + \frac{\sin C \sin A}{\sin^2 \frac{B}{2}} + \frac{\sin A \sin B}{\sin^2 \frac{C}{2}} \geq 9$ ,

where A, B, C are the angles of a triangle.

Thanks!

Darij

Source: InternLM Lean-Workbook, record lean_workbook_16299 (Apache-2.0). Complete source proposition preserved; proof developed independently.